Prosolvable group

In mathematics, more precisely in algebra, a '''prosolvable group''' (less common: '''prosoluble group''') is a group that is isomorphic to the inverse limit of an inverse system of solvable groups. Equivalently, a group is called '''prosolvable''', if, viewed as a topological group, every open neighborhood of the identity contains a normal subgroup whose corresponding quotient group is a solvable group.

== Examples == * Let ''p'' be a prime, and denote the field of p-adic numbers, as usual, by <math>\mathbf{Q}_p</math>. Then the Galois group <math>\text{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)</math>, where <math>\overline{\mathbf{Q}}_p</math> denotes the algebraic closure of <math>\mathbf{Q}_p</math>, is prosolvable. This follows from the fact that, for any finite Galois extension <math>L</math> of <math>\mathbf{Q}_p</math>, the Galois group <math>\text{Gal}(L/\mathbf{Q}_p)</math> can be written as semidirect product <math>\text{Gal}(L/\mathbf{Q}_p)=(R \rtimes Q) \rtimes P</math>, with <math>P</math> cyclic of order <math>f</math> for some <math>f\in\mathbf{N}</math>, <math>Q</math> cyclic of order dividing <math>p^f-1</math>, and <math>R</math> of <math>p</math>-power order. Therefore, <math>\text{Gal}(L/\mathbf{Q}_p)</math> is solvable.<ref>{{citation|last=Boston|first=Nigel|title=The Proof of Fermat's Last Theorem|year=2003|publisher=University of Wisconsin Press|location=Madison, Wisconsin, USA |url=http://psoup.math.wisc.edu/~boston/869.pdf }}</ref>

== See also == * Galois theory

==References== {{reflist}}

Category:Mathematical structures Category:Group theory Category:Number theory Category:Topology Category:Properties of groups Category:Topological groups